On the theory of discrete systems

Zwei einander ähnliche Aufgaben beim Entwurf von Abtastreglern mit beschränktem quadratischem Mittelwert des Reglerausgangssignals werden behandelt. Im ersten Fall besteht der Regler aus einem diskreten Teil und einem Halteglied, im zweiten Fall ist er aus einem diskreten und einem kontinuierlichen Teil zusammengesetzt, die gleichzeitig optimiert werden. Die beiden Regelprobleme werden zunächst zu Filterproblemen vereinfacht. Für beide Fälle wird eine ,, Wiener-Hopf-Gleichung für Abtastsysteme\" und ihre Lösung hergeleitet. Ein Vergleich der beiden Entwurfsverfahren wird durchgeführt.

Anwendung der Wiener-Filtertheorie zum Entwurf von Abtastreglern mit beschränkter Stelleistung

149 noise with the power spectral densit!. of (22). Using the relationship between the poner spectral densities of the input and the output of a filter a&) = €!(i)H(~-l)a&) (23) n e arrive at the expression for the desired filter, The equivalent difference equation is identical to (14). Given a sequence of : 1 7 values of zero-mean white noise xn(l <?z<:V), and an initial value of the output yu, it is possible to generate the next 2 1 ' values of the output yn(l<n<ll.'). For a zero-mean input the output will have zero mean. If a biased output is desired, the bias B must be added to each value of the output, y.. Equation (14) then becomes This equation yields an exponentially auto-correlated random sequence with standard deviation A , inverse autocorrelation time a , and mean B. in the preparation of this note.

http://ieeexplore.ieee.org/iel5/9/24089/01098226.pdf

Discussion of "Comments on the statistical design of linear sampled-data feedback systems"

A numerical method is outlined for synthesizing optimum impulse compensators in the mean-square error sense for feedback systems containing a coarse quantization actuator element where the \sigma/q ratio must be specified not only to ensure a valid estimate of the quantization error but also to limit the delay errors caused by the finite rate of actuator operation.

http://ieeexplore.ieee.org/iel5/9/24148/01100966.pdf

The design of stochastic sampled data control systems containing a discrete actuator

constraint on the estimated state x-ector p which has been overlooked. Because the first hr elements of p(a) are the output, at R successive sampling instants, of a dynamic system described by a K t h order difference equation [fl) of the original paper], each is completely determined by the values of the K preceding outputs, the K preceding inputs, and the current input. These 2K+1 values for each of the output variables in g l J are contained in p(a-lj and the last K elements of Y (~ J. Therefore, while pI1! can be selected arbitrarily to minimize the criterion D , oitly the last K eIements of ~ (~ 1 (5 + 1 j can be so chosen. But the procedure described by Koop-mans2 require? that p'\": be constrained o d y bq' the equation Hence. (1) of the paper, Koopmans' result of minimization with respect to % (a) , is inap-plicable; the vectors a;*: and the corresponding D are not correct unless the uncon-strained minimum of (4) happens to coincide with the constrained minimum. The additional constraint can, of course, be ignored if the nono\\-erlapping observation sets 3.'\"' are chosen to be so widely separated that LCa' is independent of 3.Ce-lj. However, this choice nleans a much longer period of obserx-ation to obtain an estimate with a given x~ariance, and implies a n a priori assumption about the effective settling time of the system. The proposed estimate, while apparently not optimal! may still be useful if the uncon-strained minimum is close to the constrained (true) minimum. One way of checking this possibility for a particular set of observations would be to compute Koopmans' optimal state wctor after finding the estimate ?. These vectors ai*' specif\\-a n estimated set of inputs and outputs c (i) and i(i) which can be substituted into (1) along with the a's and ,a's specified by T. I t is correct, as L. E. hIcBride, Jr. observes , that in Section I\\-of the paper referred to,l the maximum likelihood estimates were determined xithout taking into account the linear constraints bettveen elements of adjacent vectors. This was not overlooked by the author, but apparently the discussions of this point in Sections IY and \\:I1 require amplification. The estimates of Section TI-utilize only a part of the information available from the .v(?t) and y (a) sequences to estimate the pulse transfer function coefficient 1-ector y, …

https://ieeexplore.ieee.org/iel5/9/24086/01098098.pdf

J. Tou

Papers

Comments on the statistical design of linear sampled-data feedback systems